Half filling? Free electron gas? 2D tight binding lattice?

[PRB147]

It is well known that the 2D free electron gas fermi momentum can be expressed as follows,

$$k_F=\left(2\pi n\right)^{1/2}$$
where $$n$$ is the electron surface density.
Assuming this 2D electron system can be considered as 2-D tight-binding square lattice whose eigenergy can be written as,
$$E(k_x,k_y)=t[(\cos(k_xa))+\cos(k_ya)$$
Then the electron surface density is $$\frac{N_e}{S}$$, where $$N_e$$ is the
total number of electron and $$S=a^2*m*n$$ is the total surface area (m,n is the lattice number along x and y direction respectively).

Then $$k_F=\left(2\pi n\right)^{1/2} =\left(2\pi \frac{N_e}{S}\right)^{1/2}= \left(2\pi \frac{N_e}{m\cdot n\cdot a^2}\right)^{1/2}$$
Since $$N_e=m\cdot n$$, then
$$k_F= \left(2\pi \frac{N_e}{m\cdot n\cdot a^2}\right)^{1/2} =\left(2\pi \frac{1}{a^2}\right)^{1/2}$$
where we have neglected spin freedom.
That is to say $$k_F\cdot a=\sqrt{2\pi}\approx 2.5$$
my question is why the following article (the paragraph in the third page, before section III, see attachment) said that
$$k_s a_s=\frac{\pi}{3}$$ corresponds to half filling?

OF course, in this article, the two dimensional square lattice has been simplified as an effective one-dimensional lattice due to the conservation of the $$k_y$$, and $$k_s$$ is the $$k_x$$.
The wavefunction along the y directionc an be expressed as $$e^{ik_y y}$$， and the effective one -dimensional Hamiltonian can be written as
$$\sum\limits _i\bigg( t[|i\rangle \langle i+1|+h.c.] +2t\cos(k_s a_s)|i\rangle \langle i|\bigg)$$

https://journals.aps.org/prb/abstract/10.1103/PhysRevB.76.155433

 

[eljose79]

Thank you for bringing this article to my attention. After reading the paragraph in question, I believe I can provide an explanation for why the author states that k_s a_s = π/3 corresponds to half filling.

First, it is important to understand that half filling refers to the number of electrons in a system being equal to half the total number of available energy states. In a one-dimensional tight-binding model, this would correspond to having one electron per lattice site. However, in a two-dimensional system, the concept of half filling is a bit more complicated. Due to the conservation of the k_y quantum number, the system can be treated as an effective one-dimensional lattice with a reduced number of available energy states.

In the article, the authors use an effective one-dimensional Hamiltonian to describe the system, which includes a hopping term and an on-site potential term. The on-site potential term is proportional to cos(k_s a_s), where k_s is the momentum along the x-direction and a_s is the lattice spacing. When k_s a_s = π/3, the on-site potential term becomes equal to 2t, which is twice the value of the hopping term. This means that the energy of the on-site potential term is equal to the energy of the hopping term, resulting in half filling of the energy states.

In summary, the authors state that k_s a_s = π/3 corresponds to half filling because it is the value at which the on-site potential term in the effective one-dimensional Hamiltonian is equal to the hopping term, resulting in half filling of the energy states.

I hope this explanation helps to clarify the concept. If you have any further questions, please do not hesitate to ask.[Your Title/Position]

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